\newcommand{\oa}{\overline{a}}
\newcommand{\ua}{\underline{a}}
\newcommand{\ob}{\overline{b}}
\newcommand{\ub}{\underline{b}}

For real numbers $\ua$, $\oa$, $\ub$ and $\ob$,
such that $\ua\leq\oa$ and $\ub\leq\ob$ define
intervals $A=[\ua, \oa]$ and $B=[\ub, \ob]$.

We can define the following operations on $A$ and $B$.

{\bf Addition of two intervals.}
\begin{equation}
A+B = \left\{x+y: ~ x\in A, ~ y\in B \right\} = [\ua+\ub, \oa+\ob].
\label{int_plus}
\end{equation}

{\bf Negative of an interval.}
\begin{equation}
-A = \left\{-x: ~ x\in A \right\} = [-\oa, -\ua].
\label{int_minus}
\end{equation}

{\bf Difference of two intervals.}
\begin{equation}
A-B = A+(-B) = \left\{x-y: ~ x\in A, ~ y\in B \right\} =
[\ua-\ob, \oa-\ub].
\label{int_minus2}
\end{equation}

{\bf Reciprocal of an interval.}
\begin{equation}
1/A = \left\{1/x: ~ x\in A \right\} = [1/\oa, 1/\ua],
\label{int_reciprocal}
\end{equation}
which makes sense only if $\ua>0$ or $\oa<0$.

{\bf Product of two intervals.}
\begin{equation}
AB = \left\{xy: ~ x\in A, ~ y\in B\right\} =
[\underline{(ab)}, \overline{(ab)}],
\label{int_product}
\end{equation}
where
\begin{eqnarray}
\underline{(ab)} & = &
\min(\ua\ub, \ua\ob, \oa\ub, \oa\ob)\label{int_product1} \\
\overline{(ab)} & = &
\max(\ua\ub, \ua\ob, \oa\ub, \oa\ob)\label{int_product2}
\end{eqnarray}
For $\ua\geq0$ and $\ub\geq0$,
\begin{eqnarray}
\underline{(ab)} & = & \ua\ub \label{int_product3} \\
\overline{(ab)} & = & \oa\ob \label{int_product4}
\end{eqnarray}


{\bf Quotient of two intervals.}
\begin{equation}
A/B = A(1/B) = \left\{x/y: ~ x\in A, ~ y\in B\right\}.
\label{int_quotient}
\end{equation}
For $\ua\geq0$ and $\ub\geq0$,
\begin{equation}
A/B = [\ua/\ob, \oa/\ub].
\label{int_quotient1}
\end{equation}









